In an exact relativistic theory, faster than light communication is impossible (see Demystifier's post #32 and vanhees71's post #33).

However, we can also find non-exact relativistic theories, where the Lorentz symmetry is emergent at low energies. In these cases, faster than light communication is in principle possible, but very difficult in practice: https://en.wikipedia.org/wiki/Lieb-Robinson_bounds.

This is an excellent example for the discussion we have here.

First of all there is not even a mathematical proof within relativistic QFT (assuming the usual set of symmetry, causality, and stability assumptions) for the masslessness of the photon. Even when restricting oneself to local gauge symmetry and renormalizable models in the case of an Abelian gauge group, nothing prevents one from giving the photon a mass (even without using the Higgs mechanism!). On the other hand, assuming the masslessness of the photon together with the other constraints on a relativistic QFT and Wigner's analysis of the representation theory of the proper orthochronous Lorentz group leads necessary to the idea of local gauge invariance.

As you see, already into the model building itself a lot of empirical input is needed to constrain the plethora of possibilities, and among other things the masslessness of the photon is such an empirical input. Of course, one has to test this assumption to ever higher accuracy to make sure that it is really describing Nature. There's of course never 100% accuracy, and in the case of the photon mass we have only an upper limit (although an amazingly low one of ##m_{\gamma}<10^{-18} \mathrm{eV}/c^2##). So far there's no evidence for a finite photon mass and thus we set it happily to 0 in the Standard Model, as well in its classical limit, i.e., Maxwell's classical electrodynamics.

Semantic debates are so tedious and unnecessary. As in court, proof can mean sufficient evidence to convince. In mathematics (such as geometry) a proof means something very different.

@Dadface, you should have known better to say "seems to prove" (e.g. the courtroom sense) in a conversation about mathematical proof.

@ISamson , you started the confusion. In your first sentence, you mean proof in the courtroom sense (i.e. a convincing mass of experimental evidence.) In the second sentence, "the proof" can only be interpreted in the mathematical sense. You used (perhaps unintentionally) two different meanings of proof in consecutive sentences. tsk tsk. If you had said, "Is there any experimental evidence for this? I would be interested in the evidence." Then this debate would have been avoided.

Take Euclidean Geometry. If you accept its axioms then you have proved its conclusions. Everyday experience shows its axioms to a high degree of accuracy are correct - but very accurate measurements show they are wrong. But for surveyors etc the differences are so small for all practical purposes you have proved it.

What do you call that situation - beats me - I just call it science: